Microphysical Processes in the UTLS

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Microphysical Processes in the UTLS

• KEY 11

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Recommended reading

• Pruppacher and Klett, Microphysics of clouds and
  precipitation. Contains almost everything.
• Fletcher, Physics of Rainclouds (my favourite,
  albeit old).
• Young, Microphysical Processes in Clouds.
• Atkins, Physical Chemistry.

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Contents

• Classical nucleation theory (basics)
• Koops theory of water activity controlled
  homogeneous freezing of aqueous solution droplets
• Some issues with Koops theory
• Heterogeneous nucleation
• Ice supersaturation within clouds
• Volume vs. surface dominated homogeneous
  nucleation

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Some outstanding problems(see Cantrell and
Heymsfield, BAMS, June 2005)

• Homogeneous nucleation
• what role do collective fluctuations in water
  play?
• is freezing only a function of the water
  activity?
• what is the structure of the ice embryo and where
  does it form?
• Heterogeneous nucleation
• what are the most important properties of the
  heterogeneous IN?
• what are the mechanisms underlying contact and
  evaporation nucleation?
• what role do organic compounds play in ice
  nucleation?

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Classical nucleation theory
?G(r) ? (4?/3) r3 nLkT ln(e/e) 4?r2?
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Classical nucleation theory

• Number of critical nuclei Boltzmann distribution
• N(r)N0 exp(??G(r)/kT)
• ?G(r) energy required to form a critical
  nucleus
• ?G(r) 16??3/3(nLkT ln(e/e)2 (4?/3) r2?
• e, e saturation vapour pressure, actual vapour
  pressure
• ? surface tension or interfacial energy between
  droplet and vapour
• nL number concentration of water molecules in
  the liquid
• r radius of a critical germ
• Note the strong dependence on surface tension and
  temperature.

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Classical nucleation theory, contd

• Nucleation rate rate at which critical germs
  are impinged by single molecules (or larger
  clusters) to form supercritical clusters.
• J B N0 exp(??G(r)/kT),
• where BN0 is of the order 1025 cm-3sec-1.
• An accurate value of B is not really required
  since the process is controlled totally by the
  exponential function.
• Note the even stronger dependence of J on T!

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Classical theory of homogeneous freezing

• Similarly as before
• ?G(r) 16??SL3?/3nSkT ln(1/awi)2
  16??SL3?/3(?S ?T)2
• with geometrical factor ?,
• entropy of fusion per unit volume of ice ?S
• supercooling of the liquid ?T
• awieliq/eice.
• Here B(kt/h) exp(-?g/kT) with activation energy
  g for self-diffusion,
• hence
• J ? (nLkT/h) exp(-?g/kT) exp (??G(r)/kT)

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Supercooling and freezing of pure water

• Water can be cooled below its equilibrium melting
  point Tm.
• Supercooled water is in a metastable state.
• The maximum possible supercooling (Tf) can be
  achieved when the water is free of any solid
  particles that can catalyse ice germ formation.
• At about Tf freezing happens as a kinetic (i.e.
  stochastic) nucleation process, homogeneous
  nucleation.
• Heterogeneous nucleation occurs at TgtTf, actual
  temperature depends on properties of the solid
  particles.
• Tf is a genuine property of the liquid water
  alone (not classically).
• For pure water, arranged in µ-sized droplets, Tf
  is about 235 K.
• When supercooled water is in equilibrium with its
  vapour, the vapour must have 100 RH (wrt liquid
  water).
• Solution droplets have both lower Tm and lower Tf
  than pure water.

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Concept of homogeneous nucleation in the UTLS

• pure water cannot exist at Tlt-38C (supercooling
  limit)
• ice formation via homogeneous freezing of
  solution droplets
• foreign molecules (e.g. H2SO4) in the droplets
  impede formation of ice lattice
• ... until droplets are grown to sufficient size
  in supersaturated air (rarefies the foreign
  molecules)
• hence solution droplets freeze at RHigt140 (this
  threshold increasing with decreasing T)
• freezing threshold independent of chemical
  composition of the droplets
• homogeneous nucleation is driven by
  thermodynamics
• if homogeneous nucleation is the prevalent
  pathway to cirrus, then high ice-supersaturation
  must exist in the clear and cloudy atmosphere

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Solutions melting and freezing points

• Rasmussen suggested a linear relation between
  melting point depression and supercooling
  required for homogeneous nucleation of aqueous
  solution droplets
• ?Tf ? ?Tm
• ?Tf Tf0Tf
• Tf0 is the supercooling limit of pure water (235
  K),
• ?Tm Tm0Tm
• Tm0 273.15 K.
• The constant ? is independent of concentration,
  but depends on the chemical composition of the
  solute
• ? is an empirical constant and cannot be derived
  from first principles.

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Classical treatment of solution freezing

• Jsolution(T) Jpure water(T ?Tf)

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Water activity-based nucleation theory (1)
Koop, 2004
When melting and freezing temperatures of water
and various aqueous solutions are plotted vs.
water activity, the data collapse on two single
curves, with little scatter in the case of the
freezing temperature. This implies that
homogeneous freezing is independent of the
chemical nature of the solute.
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Water activity-based nucleation theory (2)
water activity aw saturation vapour pressure
over solution saturation vapour pressure over
pure water the vapour pressure over the
solution equals that of pure ice at the melting
temperature Tm eice (Tm) eliq (Tm) aw or
awi (Tm) aw Tf (aw) Tm (aw - ? aw) ? aw
independent of chemistry. ? aw 1 - awi (Tsc,
max) 0.305
The locus of the Tf curve is probably a
determined by the perturbation to the hydrogen
bonding network induced by the foreign molecules.
(Koop 2004)
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Water activity-based nucleation theory (3)
saturation vapour pressure over water
saturation vapour pressure over ice e liq (Tm)
/ e ice (Tm) (upper inverse aw scale)
melting curve (lower aw scale) eice (Tm) /
eliq (Tm) awi
critical supersaturation for homogeneous
freezing (upper inverse aw scale), red curve ?
blue curve e liq (Tm) / e ice (Tm) ? aw (Tf)
freezing curve
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Water activity-based nucleation theory (4)
water saturation
critical supersaturation
fit Si,crit 2.352 - 3.883?10-3 T J 5.5 ?109
cm-3 s-1 e-folding freezing time 43 s for 1 µ
droplets, or 1 s for 3.5 µ droplets
Threshold supersaturation for homogeneous
nucleation increases with decreasing temperature
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Why does water activity control hom. freezing ?

• Solutes affect the equilibrium and
  non-equilibrium properties of water substance.
• Ice nucleation is affected by the solute
  molecules,
• increasing solute concentration ? increasing
  supercooling necessary for freezing.
• Peculiar properties of supercooled water
• interactions between water molecules via hygrogen
  bonds.
• nearly tetrahedal arrangement of the two H atoms
  and the two free electron pairs around the
  central O atom
• preference of tetrahedral co-ordination in the
  local water structure.
• Mechanical pressure and foreign (solute)
  molecules change the preferred interatomic
  distances, hence the water structure.

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How can the state of waters hydrogen bonding
network define the locus of the Tf curve?

• There are several theories
• Stability limit theory (Rasmussen and coworkers)
• Proximity of the freezing curve to a postulated
  stability limit bounding a region where
  isothermal compressibility is positive.
• The singularity-free scenario (Archer and Carter)
• Existence of a second critical point (Baker and
  Baker)

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Theory of the 2nd critical point

• initiation of freezing in pure water
• liquid compressibility and density fluctuations
  reach maxima.
• Temperature of the onset of freezing is an
  equilibrium property of the liquid phase alone.
  (Remember strong influence of surface tensions in
  classical theory. Not so here!)
• Analytic model of liquid water thermodynamic
  response functions have extrema at atmospheric
  pressure and 235 K.
• predominance of weak H-bonds at higher
  temperatures
• predominance of strong H-bonds at lower
  temperatures
• locus of the extrema is a region of a significant
  change in the character of the H-bonding network
• loci of the compressibility maxima and the
  freezing curve are nearly the same at atmospheric
  pressures.

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Heuristic argument and However..

• As T approaches Tf density fluctuations rise. So
  the probability rises to find in the liquid
  regions where the density approaches that of ice.
  
• However, as a function of T at atmospheric
  pressure the extrema of compressibility etc. are
  much weaker then the sharpness of the sudden
  increase of the freezing rate. This makes this
  explanation somewhat unconvincing.
• See Baker and Baker, GRL, 2004

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test of Koops theory in the AIDA cloud chamber
Good agreement between measurements and model
results show that Koops parameterisation is able
to predict correctly homogeneous nucleation of
H2SO4/H2O solutions in the AIDA chamber.
Non-equilibrium effects lead to slightly higher
critical supersaturations as in Koops
equilibrium theory.
Haag et al., ACP, 2003
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Some issues with Koops theory

• derivation of Tf(aw) needs assumption that aw
  does not depend on temperature. This is indeed
  often the case above Tm where the water activity
  can easily be measured.
• Below Tm, aw must often be determined using
  models or extrapolations.
• For sulphuric acid, aw is nearly T-independent.
• However, there are exceptions, e.g. ammonium
  nitrate NH4NO3.
• aw(NH4NO3) increases with decreasing T
• decreasing interaction of NH4NO3 with H2O at
  lower T
• more and more ion-ion recombination (NH4 with
  NO3?), which makes it invisible for the water
  molecules.
• decreasing solubility of ammonium nitrate in
  water upon cooling.

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issues with Koops theory ammonium sulfate
behaves differently
from Paul DeMott
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measurements of very large supersaturation,
influence of organics?
no organics
with organics
Jensen et al., ACP, 2005, report on measurements
of very large supersaturations in very cold air,
Si being much larger than Si,crit
De Mott et al., PNAS, 2003
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Kärcher Koop, 2005, show that organic material
within the solution droplets is able to impede
nucleation such that the peak supersaturation can
be much higher than Si,crit
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possible impedence of freezing by organic
materials or surfactants
Kärcher Koop, 2005

• At a certain temperature it needs a certain
  activity for freezing (big dots).
• Different solutions reach that activity at
  different solute mass fractions (W).
• Solutions containing organics generally have
  higher solute mass fractions than inorganic
  solutions when the critical activity is reached.
• This implies
• less water,
• smaller particle volume (?1/W),
• smaller freezing nucleation rates.

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freezing temperature and nucleation rates

• Koops theory is able to predict freezing
  temperatures or critical supersaturation for
  homogeneous freezing.
• it is not able to predict freezing rates.
• Freezing rates are parameterised in Koops paper
  as a function of aw?awi.
• In the classical theory it is relatively
  straightforward to envisage
• critical germ
• attack frequency by single molecules
• ? nucleation rate.
• The notion of an ice germ does not exist in
  Koops theory.
• Hence difficult to see how a nucleation rate
  could be derived within the framework of this
  theory.

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heterogeneous nucleation

• Classical framework
• Energy for germ formation (contact angle ?)
• ?G(het) ?G(hom) f(cos ?) with 0 ? f(cos ?)
  ? 1
• ? ?G(het) ? ?G(hom),
• i.e.
• lower critical supersaturation or higher critical
  temperatures (less supercooling) for
  heterogeneous nucleation.

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heterogeneous nucleation

• Water activity based framework
• Zuberi at al. 2002
• it may be possible to compute freezing
  temperatures for solution droplets with insoluble
  inclusions in a way analogous to the description
  of homogeneous nucleation by Koop et al. 2000.
• However, the scatter in the measured freezing
  temperature in the aw-T diagram is large and the
  fit is not perfect.
• It could be that such an analogy is indeed
  there, but if there is not a single value of ?aw
  (a single value of maximum supercooling of pure
  water drops with insoluble inclusions) such an
  analogy does not help much.

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Measurements of het. freezing in the AIDA chamber
The AIDA chamber at IMK in Karlsruhe is a large
(84 m3) cloud chamber. Freezing is initiated by
quasi- adiabatic expansion. Ice crystals appear
and start to grow as soon as the critical
supersaturation characteristic for the IN is
reached. Different species have different
thresholds.
See Stefanie Schlichts poster!
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observations of nucleation thresholds in data of
RHi
ambient RHi
in cloud RHi
Haag and Kärcher, 2003
onset of heterogeneous freezing
onset of homogeneous freezing
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Ice supersaturation within clouds

• The usual thinking is that after a (short) while
  the relative humidity in a cloud should approach
  saturation. However, this while, the so-called
  relaxation time, can last very long, depending on
  temperature and crystal number concentration.
• ? g (4?/3) N D(T,p)-1
• When the updraught goes on after cloud formation,
  saturation is not reached because of the ongoing
  decrease of the saturation pressure. Instead a
  residual supersaturation of a few percent will be
  the stable situation.
• sasympt. ?g/(?u? ?g)
• with updraft time scale ?u (Rv cp T2) / (Lgw),
  provided ?u gt ?g

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• Sometimes the relaxation time is longer than
  other relevant time scales within a cloud, e.g.
  the sedimentation time scale. Then saturation
  will never be reached within a cloud.

Altitude (m)
Ni 5L-1, w 4.5 cm/s, RHihet 130
Time (min)
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Ice supersaturation within clouds examples
Comstock et al. (ARM data)
Ovarlez et al. (INCA data)
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All CRYSTAL-FACE RHi within clouds
Physical-chemical effects may also cause
persistent supersaturation within clouds, e.g.
Delta-ice or cubic ice.
Mean RHi binned by T open circles. 13 July 2002
contrail triangles. 19 July 2002 contrail
diamonds. (taken from Figure 1 of R. S. Gao et
al., Science, 2004)
From R. Herman, 2004
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Where does homogeneous nucleation occur?Drop
Volume or Surface?

• Traditionally the homogeneous nucleation process
  is described as a process that occurs somewhere
  in the bulk of the droplet that then freezes
  completely.
• Freezing rate is then proportional to the droplet
  volume
• Freezing of an ensemble obeys
• Punfrozen at time t exp(-JVt) with J
  cm-3s-1
• Djikaev et al. (JPCA, 2002) and Tabazadeh et al.
  (PNAS, 2002) have found indications and given
  arguments that homogeneous nucleation (germ
  formation) should instead proceed close to the
  droplet surface.

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wetting criterion

• Condition at least one facet of the crystal is
  only partly wetted by liquid water (could be
  valid for the ice-water system).
• ?vs? ?vl lt ?ls
• surface energies extrapolated to T ?40C
• ?vs 102 to 111 mJ/m2
• ?vl ? 87 mJ/m2
• ?ls 15 to 25 mJ/m2
• Hence ?vs? ?vl is approximately in the range 15
  to 26 mJ/m2.
• The surface energies are only measured for
  systems with macroscopic dimensions, not for the
  small clusters containing only some tens of
  molecules.

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If homogeneous nucleation is a surface process,
freezing experiments with droplets in
oil-emulsions can be affected by the oil. Is
this also relevant for droplets in an atmospheric
environment?
Tabazadeh et al., PNAS, 2002
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• Duft and Leisner (ACP, 2004)
• Test of the surface nucleation hypothesis using
  an electrodynamic
• droplet levitation apparatus.
• droplets of 19 and 49 µm radius have the same
  (volume-) nucleation rates.
• At least for such large droplets freezing seems
  to occur preferentially in the bulk.
• Relevant for freezing of droplets in Cb or fog.
• Experiments do not exclude that in sub-micron
  droplets ice germs form preferentially at the
  surface.